Leslie,
A few thoughts on your proposed travel plans....
The first thing to keep in mind is that you'll arrive in Europe the day after you departed. You'll most likely be tired and jet-lagged when you arrive at Milano/MXP, and it's going to take at least several hours to get to your Hotel in Venice.
You'll likely land at MXP Terminal 1 and your best option for travel to Milano Centrale is one of the two airport Bus Shuttle services. Be prepared for the chaos at Milano Centrale and be vigilant with your luggage and valuables!
You can buy your Rail tickets (and reservations) to Venezia Santa Lucia in Milan (note there are 2 stations in Venice). The ticket office is two floors down from the tracks, and there's often quite a queue. The automated machines are a good way to avoid the crowd. The machines on the main floor accept only credit cards, however those down by the ticket office will accept Euro's as well (and are not as busy as the main floor machines).
The rail trip to Venice will be about 2.5 hours. When you arrive, walk out the doors and down the steps to the Vaporetto stop. The Italy Guidebook has good information on which boat to take and the costs.
My suggestion for the order of cities would be Venice-Florence-Cinque Terre-Rome. There's a direct train from La Spezia (CT) to Roma Termini (about 4 hours, reservations compulsory).
Regarding Hotels, my preference would be to pre-book all of them. July is PEAK travel season in Europe and you may find that many are already fully booked. Pre-booking allows a better choice of Hotels, both in terms of price range as well as which area of each city to stay in (that way you won't end up in a dodgy part of town).
How long will you have after your friend departs? A trip to the Amalfi coast is certainly possible, however keep in mind that the culture gets more "intense" as you travel south of Rome.
I'm assuming you'rre flying home from Rome?
Cheers!